Linearized instability analysis of frame structures under ...

Coupled Systems Mechanics, Vol. 10, No. 1 (2021) 79-102

DOI: https://doi.org/10.12989/csm.2021.10.1.079 79

Copyright © 2021 Techno-Press, Ltd. http://www.techno-press.org/?journal=csm&subpage=8 ISSN: 2234-2184 (Print), 2234-2192 (Online)

Linearized instability analysis of frame structures under non-conservative loads: Static and dynamic approach

Emina Hajdo1, Rosa Adela Mejia-Nava2, Ismar Imamovic1

and Adnan Ibrahimbegovic2,3

1Faculty of Civil Engineering, University of Sarajevo,

Patriotske lige 30, Sarajevo, BiH, Bosnia and Herzegovina 2Université de Technologie Compiégne, Laboratoire Roberval de Mécanique,

Rue du Dr Schweitzer, 60200 Compiègne, France 3Institut Universitaire de France, France

(Received November 27, 2020, Revised December 12, 2020, Accepted January 18, 2021)

Abstract. In this paper we deal with instability problems of structures under nonconservative loading. It is shown that such class of problems should be analyzed in dynamics framework. Next to analytic solutions, provided for several simple problems, we show how to obtain the numerical solutions to more complex problems in efficient manner by using the finite element method. In particular, the numerical solution is obtained by using a modified Euler-Bernoulli beam finite element that includes the von Karman (virtual) strain in order to capture linearized instabilities (or Euler buckling). We next generalize the numerical solution to instability problems that include shear deformation by using the Timoshenko beam finite element. The proposed numerical beam models are validated against the corresponding analytic solutions.

Keywords: instability problems; non-conservative load; Euler-Bernoulli beam; von Karman strain;

Timoshenko beam; shear deformation

1. Introduction

Modern urban architecture trends are geared towards constructing tall structures, with either

office/residential buildings or transport infrastructure. At the same time, the progress in material

engineering that allows to increase the load-bearing capacity and safety of the structure, results with

lightweight and slender structural elements for high-rise structures, which all became a symbol of

modern engineering. However, by pushing the boundaries of engineering in the aforementioned way,

and by designing slender and lightweight structures can result in fatal consequences of loss in

structural stability. Namely, the instability phenomena imply that for the reduced structural stiffness

at so-called critical equilibrium state, even a small perturbation can lead to a disproportional increase

in response (displacement, strain and stress) and eventual structural failures (e.g., see illustrative

damage cases for recent US storm Laura url: https://www.fox23.com/news/photos-

hurricane/KCGRTCQ4LFFNDHZLW2O6LEE26I/). Thus, the analysis of complex structures

Corresponding author, Assistant Professor, E-mail: [email protected]

Emina Hajdo, Rosa Adela Mejia-Nava, Ismar Imamovic and Adnan Ibrahimbegovic

instability, and adequate prevention of its occurrence, has again become one of the fundamental

problems faced by engineers. Furthermore, different types of structures, next to (permanent)

gravitational forces, are also subjected to various loads. In general, all the forces that act upon the

structure can be classified in two categories: conservative and non-conservative. For the

conservative forces (e.g., gravitational force), the external work does not depend on the path but

only on the starting and ending point. Therefore, the conservative force can be obtained from a

potential (e.g., gravity potential) and thus the total work done by a conservative force on a closed

path is equal to zero. The forces attributed to external actions (e.g., wind or fluid flow, or yet a

frictional force) are often non-conservative (Sugiyama et al. 2019). The work of nonconservative

forces cannot be computed from a potential, and it depends on the path taken. It is a fundamental

finding of Bolotin (1964) that the structure instability under non-conservative forces may occur

dynamically, leading to increasing vibration amplitudes (yet called flutter instability). Therefore, the

problem of structural instability under non-conservative forces should be examined within the

dynamics framework.

Vast majority of analytic solutions is already available (Hajdo et al. 2020, Bolotin 1963,

Lacarbonara and Yabuno 2006, Jeronen and Kouhia 2015) for the classical problems of a column

buckling under non-conservative load. This kind of non-conservative force is also called a follower

force (Timoshenko and Gere 1961, Langthjem and Sugiyama 2000, Beck 1952, Farhat et al. 2013,

McHugh and Dowell 2020). Follower forces depend on the deformation of structures. Namely, these

forces follow the structure motion (e.g., cross-section rotation), and are thus path-dependent and not

possible to derive from a potential (Sugiyama et al. 2019). The stability analysis of such a column

under follower force cannot be obtained by using the statics approach of studying the critical

equilibrium state (Ibrahimbegovic et al. 2013, Hajdo et al. 2020, Imamovic et al. 2019), but requires

using the dynamics framework. Moreover, the stability analysis of such column cannot be placed

within the classical Lyapunov stability framework of dynamic motion (Mejia Nava et al. 2020,

Lozano et al. 2000), which pertains to variations of the total energy (Lyapunov functional in

Mechanics), simply because no such potential can be established for non-conservative forces.

We propose in this work an original approach of “dynamic” equilibrium equations, where the

structure stability can be confirmed by proportionally small vibrations due to small perturbations

from its equilibrium position. Such theory of the dynamic instability can be explained by using an

example of the column under compressive axial follower force. If such a force is smaller than the

Bolotin critical force value, the column response will remain proportionally small due to a small

perturbation. However, for the applied force equal or larger than critical, relation between a small

perturbation and structure response will become disproportionally large, followed by large vibration

amplitudes, which are rapidly increasing (Bolotin 1964). A previous study of Bolotin (1963) was the

first to give the analytic solution to elastic column instability problem under non-conservative axial

loads. Similar studies of simple structures can also be found in (Timoshenko and Gere 1961), or in

a recent survey paper (Langthjem and Sugiyama 2000), for different kinds of columns (Beck’s,

Reut’s etc.) under non-conservative follower loads. These and other works (Jeronen and Kouhia

2015, Beck 1952, Farhat et al. 2013, McHugh and Dowell 2020), exploring effects of damping,

flutter instability of aircraft wings or columns subjected to rocket thrust, mostly put the emphasis on

either analytic solutions or their experimental validations. The study of large beam deflections under

conservative and non-conservative load, in static framework can be found in (e.g., Masjedi and

Ovesy 2015). Dynamic instability of geometrically exact beams under a follower force using fully

intrinsic equations was subject of research in (e.g., Amoozgar and Shahverdi 2016). Many

researchers studied follower force as a distributed load (e.g., Fazelzadeh et al. 2017). A uniform

80

Linearized instability analysis of frame structures under non-conservative loads…

cantilever beam subjected to a partially follower load and analyzed using finite element method is

presented in (Gasparini et al. 1994). A detailed review of papers that deal with the topic of dynamic

stability of the structures under follower force can be found in (Elishakoff 2005). The main thrust in

this work is directed towards a complementary approach based upon numerical modeling by using

the finite element method (Ibrahimbegovic 2009).

In particular, as the first novelty, we introduce the Euler-Bernoulli beam finite element that can

deal with instability problems under non-conservative follower force. This is achieved by using the

von Karman virtual strain measure, typical of moderate rotations, which can account for instability

phenomena within the framework of weak form of instability problem. Some previous studies

included the von Karman virtual strain to capture the buckling of frames (e.g., Dujc et al. 2010,

Piculin and Brank 2015). In our previous work (Ibrahimbegovic et al. 2013, Hajdo et al. 2020), the

von Karman strain was used to formulate the structure instability problem as corresponding linear

buckling problem, which reduces to solving the linear eigenvalue problem to predict the critical load

for conservative (quasi-static) forces. For the instability problems under non-conservative force

studied herein, we also introduce the novelty of adapting the beam element for use in the dynamics

framework, and obtain the corresponding dynamic equilibrium solution by employing the Newmark

time-integration scheme. The final novelty in this paper is in adapting the proposed approach also

to shear deformable beams, with the equivalent of Timoshenko beam model. Some background

material and more details needed to master the numerical models developed in this research can be

found in our previous work (Ibrahimbegovic et al. 2013, Hajdo et al. 2020, Imamovic et al. 2019,

Mejia Nava et al. 2020, Ibrahimbegovic 2009, Medic et al. 2013).

The outline of our paper is as follows. In Section 2, we present analytic solutions (based upon

the strong form) for a cantilever beam under fixed and follower force, as well as semi-analytic

solution for static and dynamic instability of frame structure. The discrete approximation is

constructed by FEM as shown in Section 3. In Section 4 we give several illustration examples, and

in Section 5 we state conclusions.

2. Strong form based analytic solution

In this section we briefly discuss the analytic solutions for instability of a compressed cantilever

under fixed and under follower force. The latter case was first solved by Bolotin (1963), with the

analytic solution satisfying all three fundamental equations of the strong form. We then provide the

solution for a shear deformable beam, where the Timoshenko rather than the Euler-Bernoulli beam

model was used. We finally provide an extension of such solution to instability of frame structure,

in terms of semi-analytic solution.

2.1 Cantilever beam instability under fixed load - analytic solution

The analytic solution presented in this section, based upon strong form of Euler-Bernoulli beam,

verifying the equations of kinematics, constitutive law and equilibrium. The first two are borrowed

from linear beam theory.

1. Kinematics (see Fig. 1(b))

( )( )

( ) ( )( )

2

2

p p pp

xx

p

dv xu x,y y y du d v

x,y ydxdx dx

v x,y v x

= − = −

= = −=

(1)

81


2. Constitutive law

( ) ( )

( ) ( )

2

2

2 2 232

2 2

212

p

xx xx

h

xx

A h

d vx,y E x,y Ey

dx

d v x d v xbhM y dA Eb y dy E EI

dx dx

−

= = −

= − = = =

(2)

where b and h are the cross section width and height respectively, M is a bending moment, E modulus

of elasticity and I is a moment of inertia. Eq. (2) only holds for rectangular cross-sections. However,

the last equation takes into account nonlinearity to establish equilibrium in the deformed

configuration, i.e., on the beam deformed axis:

3. Equilibrium:

( ) ( )

( )( )

( )( )

( )

( ) ( )

( )( )

( )

( ) ( )

( )( )

0

0

02

y

dT xT x dx

dx

dM xM x dx dx

dx

dT xF T x T x dx q x dx q x

dx

dx dvM x M x dx q x dx T x dx P dx

dx

dM x dvT x P

dx dx

+

+ +

= = − + + + = +

= = − + + − + +

= + +

(3)

( ) ( )( )

( ) ( ) ( )

4 2

4 2

4 2

4 2

d M x d v xP q x

dx dx

d v x d v x q xP

dx EI dx EI

+ =

+ =

(4)

were T is a shear force, q is a transverse distributed load. Thus, we are interested in the class of

problems known as linearized instability or Euler buckling, which considers small deformations

with moderate rotations before reaching the critical equilibrium state. By combining these three

equations we obtain the strong form of the problem for Euler buckling

( ) ( )4 2

2

4 20

d v x d v x Pk ; k

dx dx EI+ = = (5)

The general solution for such equation can be written as

( ) 1 2 3 4v x A sinkx A coskx A x A= + + + (6)

where A1, A2, A3 and A4 are constants to be obtained from boundary conditions.

The boundary conditions for cantilever beam in Fig. 1 can be written as

( ) ( )

( )

( )( )

( ) ( ) ( ) ( )

2

2

3 3

2

3 3

0 0 0

00 0 0

d v lv ; M l EI

dx

dv d v l dv l dv l d v l ; T l EI P k

dx dx dx dx dx

= = =

= = − − = → + =

(7)

By combining relations in Eqs. (6)-(7), we further obtain

82


(a) Initial configuration (b) Deformed beam axis

Fig. 1 Linear instability of cantilever

( )

( )

( )

( ) ( )

1

222 2

322

43

20

3

0 0

00 1 0 1 00

0 1 0 0

0 0 00

0 0 0 0

0 det

v

dvA

dxAk

d v lAk sinkl k cos kl

dxAk

dv l d v lk

dx dx• =

=

= = − −=

+ =

(8)

From the above determinant it follows that

2 2 10 0 1 2

2

nk cos kl cos kl kl , n , ,...

−= = = = (9)

If we recall the relation for k given in Eq. (5), further we get

( ) ( )2 2 2

2 2

2 2

2 1 2 1

4 4

n n EIPk P

EI l l

− −= = = (10)

Finally, the value of the critical force for Euler’s buckling of cantilever beam is

( )

2

22

cr

EIP

l

= (11)

while buckling modes are described with

( )( )2 1

12

n

n xx cos

l

−= − + (12)

The first buckling mode corresponding to the smallest value of the critical load in Eq. (11) is

( )12

xx cos

l

=− (13)

In the reminder of this section we extend this analytic solution to the case of shear deformable

beam, represented by Timoshenko beam model (see Fig. 2).

P

l

x

y

83


Fig. 2 Deformed configuration of an axially compressed Timoshenko cantilever beam

In the same manner we can now obtain the critical force value for Timoshenko beam as the

solution of the strong form combining three fundamental equations: kinematics, constitutive

equations and equilibrium. The kinematics of the Timoshenko beam is given as

d dv ;

dx dx

= = − (14)

where θ is independent rotation filed that describes the cross-section position, resulting with new

definition of bending κ and shear strain γ.

The corresponding expression for bending moment M and shear force T can be written from

constitutive equations as

( ) ( )M EI ; T GA = = (15)

where G is a shear modulus and A is a cross-sectional area. According to the Fig. 2, the equilibrium

on the deformed configuration results in

M P v ; T Psin P

= − = = (16)

Further, by combining the results obtained in Eqs. (14)-(16) we can obtain the relation between

the section rotation and beam transverse displacement

2

21

dv P

dx GA

d v P d

dx GA dx

= +

= +

(17)

With this result in hand, the first relation given in the Eq. (15) can be written

2

2

0

10

1

dEI P v

dx

d vEI Pv

Pdx

GA

+ =

+ =

+

(18)

Finally, the differential equation expressing the strong form of instability condition, of the axially

compressed Timoshenko beam is given as

2

21 0

d v P Pv

dx EI GA

+ + =

(19)

T

84


Previous equation can be rewritten in the same format as the one for Euler- Bernoulli beam but

by introducing a new definition of the parameter k

( )4 22

4 2

10

P P GAd v d v k ; kdx dx EI

++ = = (20)

The general solution of this differential equation can again be written as


where A1, A2, A3 and A4 are constants to be determined by taking into account the boundary

conditions

( ) ( ) ( ) ( )0 0 0 0v v l , M l , T l= = = = (22)

Of particular interest is in a nontrivial solution, which is obtained for the case when

2 10 1 2

2

ncos kl kl n , ,...

−= = = (23)

The smallest value of factor k corresponding to the critical force value is computed for n=1. With

the last result in hand, by using the relation given in Eq. (19), we get

2

21

2 4

crcr cr

P EIk P

l GA l

= + =

(24)

By using the last expression, the critical force for Timoshenko beam is obtained as

2

2 2

2

1

21 1

cr

EIP

l EI

l GA

=

+ +

(25)

It is obvious that the critical force value computed for Timoshenko beam is smaller than the one

obtained for the Euler beam, and when 0 → , GA→ , 2

24cr

EIP

l

→ . This finding is in agreement

with Cauchy interlace theorem (e.g., Ibrahimbegovic 2009) since the Timoshenko beam model

allows for more deformation modes than the Euler beam model.

2.2 Dynamics framework for instability problems

It was first shown by Bolotin (1964, 1963) that the same procedure no longer works when we

have the follower force. In such a case we have to place the instability problem within the dynamics

framework. Namely, we now consider a deformed configuration of a beam under compressive axial

force P, which undergoes vibrations that should be examined within the dynamics framework.

2.2.1 Cantilever beam analytic solution for thin beams by Bolotin In this section we present the solution to the problem of instability of the Euler-Bernoulli beam

with a concentrated mass on top under the action of a compressive follower force. This implies that

the force rotates together with the free end section of the beam and remains tangential to its deformed

axis (see Fig. 3).

The analysis of this problem is again carried out under hypothesis of linear kinematics and linear

85


Fig. 3 Euler-Bernoulli cantilever beam under follower load

constitutive equation

( )2 2

2

2 2;

d v d vx,y y E M EI

dx dx = − = → = (26)

The only nonlinear equation in this instability problem pertains to equilibrium, first assumed as

static, which applies in the deformed configuration (indicated by a curved solid line in Fig. 3 above).

If we assume small deflection and small end rotation with

dv

dx = (27)

the follower force can be discomposed into horizontal and vertical components with: xP P ,

yP P . The moment equation of the (slightly) bent beam is given in (Bolotin 1963)

( ) ( )2

2

d vEI P f v P l x

dx= − − − (28)

where EI is the beam bending stiffness, f=v(l) is the deflection at the free end of the beam and l its

length. We can propose the general solution of this differential equation as

( ) ( )1 2 1v x A sinkx A coskx f x= + + − − (29)

where A1 and A2 are arbitrary constants, whereas f and φ denote free end displacement and rotation.

The boundary conditions for this problem are zero displacement and zero slope at cantilever

beam support point and assumed values of displacement and slope at the free end with

( )( )

( )( )0

0 0 0dv dv l

v , , v l f , dx dx

= = = = (30)

If we substitute the general solution into these boundary conditions, we obtain the following set

of algebraic equations, written in the matrix notation:

1

2

0 1 1 1 0

0 0 1 0

0 0 0

0 0 0

A

k A

sinkl cos kl f

cos kl sinkl

− =

−

(31)

86


It is easy to check that the determinant of this matrix is equal to -1, which means that there are

no other solutions but trivial, i.e., ( ( ) 0v x ). Hence, we conclude that the static approach cannot be

used to solve this instability problem.

For that reason, we must turn towards finding the solution of such instability problem within the

dynamics framework. In other words, we study the possibility of flutter instability. Here, the

equilibrium is said to be unstable if a small disturbance causes a large deviation of the system from

the considered dynamic equilibrium. With the follower (non-conservative) load explicitly dependent

on time, the equation of motion with small oscillations around equilibrium state takes the following

form

( )( ) ( )( ) ( )( ) ( )

( )2 2

2 2

d v x,t d f tEI P f t v x,t P t l x m l x

dx dt= − − − − − (32)

where v(x,t) is the dynamic deflection at each point and f(t) is the deflection at the free-end of the

cantilever, i.e., f(t)=v(l,t). The solution to this equation can be obtained by separation of variables

( ) ( ) ( ) ( )i t i t i tv x,t x e , f t fe , t e = = = (33)

where we can assume that the beam performs free vibrations with frequency ω. We can thus reduce

the Eq. (32) to the form

( ) ( )2 2

2 2 2

2

d m fEI k k P k l x l x

dx EI

+ = − − + − (34)

The general solution of the last equation can be written as

( ) ( ) ( )2

1 2 2

m fx A sinkx A coskx f l x l x

k EI

= + + − − + − (35)

Making use of the corresponding boundary conditions for cantilever beam v(0)=0, v’(0)=0, v(l)=f,

v’(l)= , we obtain a new form of the system in Eq. (31), and again check for non-trivial solutions

by enforcing that

( ) ( )

( ) ( )

2

2

2

2

2

2

0 1

0 10

0 0

0

m l1+ l

k EI

mk

k EI

sin kl cos kl

mkcos kl ksin kl

k EI

−

− = − −

(36)

We can further obtain the corresponding value of ω from zero determinant condition in Eq. (36)

above, and find out that such value becomes infinite (non-physical) if the following condition is

verified

1P tankl kl

sinklmlcos kl

kl

= → =

−

(37)

87


The last expression provides the numerical solution for the critical value 2 05kl . = . Thus, the

critical follower load for Euler-Bernoulli beam can be computed as

2

2

2 05cr

P .k P EI

EI l

= → = (38)

If we now have a cantilever beam with the distributed mass (see Fig. 4) the beam motion is

governed by

4 2 2 4 2 22 2

4 2 2 4 2 20 0

d v d v d v d v d v d vEI P m k

dx dx dt dx dx dt+ + = + + = (39)

The boundary conditions at the fixed end and at the free end of cantilever are given as:

( )( ) ( ) ( )2 3

2 3

00

v ,t v l ,t v l ,tv ,t = 0 ; = = 0

x x x

=

(40)

Furthermore, we can rewrite the differential equation of the free vibrations of the column as

follows

4 2 22

4 20

d g d g mk g

dx dx EI

+ − = (41)

Fig. 4 Beck’s column under a follower force

The general solution of the differential equation is given with

( ) 1 2 3 4g x B sin x B cos x B sinh x B cosh x = + + + (42)

where α and β are

1 2 1 22 2

4 4

4 41 1 ; 1 1

2 2

k m k m

EIk EIk

= + + = − + +

(43)

88


By taking into account the boundary conditions, and searching for the nontrivial solution, we

obtain the critical force value as follows

2

2

2cr

EIP

l

= (44)

2.2.2 Analytic solution for Timoshenko beam buckling under follower force The instability problem for Timoshenko beam buckling can be defined in a similar manner, by

using linear kinematics and constitutive equations, combined with nonlinear equilibrium equation

imposed in Timoshenko beam deformed configuration. We have already written all governing

equations for the Timoshenko beam instability in Eqs. (14)-(24). We have shown the equivalence

of the differential equation governing the instability problem for Timoshenko beam, but with

different value of k. Hence, the solution to such instability problem will again be computed within

the dynamics framework. One can follow the same steps to arrive at the same final result as the one

in Eq. (37), but with different value of k. Such an approach can now provide the critical value of

the follower load for Timoshenko beam, by imposing that

22

2

12 05c

PP

GA .k k

EI l

+

= → = (45)

22 051 4 1

2c

cr

c

. EI

GAP

GA

+ −

= (46)

2.3 Static and dynamic instability of frame: semi-analytic solution

In this section we give an analytic solution for the critical load of a frame under two conservative

loads applied at the corner nodes. The frame geometry is as shown in Fig. 5.

Under the applied loads, we will have the deformed configuration of the beam as shown in Fig.

6(a). The main idea for constructing the solution is to propose an equivalent system that would have

the same deformed shape. Such an equivalent system consists of a cantilever beam with a rotational

spring at the free end of the beam, which is an approximate replacement for the restraining beam

Fig. 5 Frame instability under conservative load

89


(a) Deformed shape (b) Equivalent system for the frame

Fig. 6 Frame under conservative load

action upon the column (see Fig. 6(b)).

The corresponding differential equation characterizing critical equilibrium state for this problem

is similar to the one in Eq. (5) for a cantilever beam

4 22

4 20

d v d v Pk ; k

dx dx EI+ = = (47)

The general solution to this differential equation can be written as follows


The main modification from the previously developed analytic solution by Bolotin for cantilever

beam is a new form of the boundary condition of top of the column, which is brought by the presence

of the spring. Thus, we can write as follows

( )0 0v = (49)

( )0 0dv

dx= (50)

( ) ( ) ( ) ( )2 2

2 20 0

d v l dv l d v l dv lEI ; ;

dx dx dx dx EI

+ = + = = (51)

( ) ( ) ( ) ( )3 3

2 2

3 30 0

d v l dv l d v l dv l PEI P ; k ; k

dx dx dx dx EI+ = + = = (52)

where α and β are the parameters quantifying the beam bending stiffness that is retaining the

rotations θ1 or θ2 at top of each column. The value of this parameter is obtained by using the FEM

solution as shown next. For that reason, we refer this development as semi-analytic solution.

Namely, by using the finite element method solution based upon the Hermite polynomials (see

Section 3.2) representing the beam deformed shape. We note in passing that such approximation is

exact for pure bending of beam with homogeneous cross-section with all loads applied at the nodes,

but not for critical mode where deformed shape can be represented by trigonometric functions (e.g.,

Timoshenko and Gere 1961). We can obtain the following approximation of beam bending

contribution to critical mode

90


1 1

2 2

4 2 6

2 4 6

EI EI EI

M l l l

EI EIM EI

l l l

= =

(53)

which indicates the correct value for parameters 6EI l = and b =a EI = 6.

With these boundary conditions we can obtain the critical load by following the usual procedure

and enforcing the cero the value of the corresponding determinant

1

2 2

3

42 2

0

0 1 0 10

0 1 00

0 0 00

0 0

det

Ak

Ak

A( k cos kl k sin kl

Ak sin kl ) k cos kl )

=

=

(− − −

(54)

which allows us to obtain the critical load solution as follows

( )2

2 2

2 71 7 344cr

, EIP , EIk P

EI l l= → = = (55)

We can see that such a semi-analytic solution is slightly lower than the critical load that we obtain

with the FEM approximation of deformed shape by Hermite polynomials used the whole frame

(beam and columns); which is equal to

2

7 5cr

, EIP

l= (56)

That is why we seek the analytic solution wherever possible to get the best estimate of the critical

load and to validate our FEM solutions.

In numerical simulation to follow, we propose three different frames (see Fig. 7) each loaded

with two compressive conservative or non-conservative loads. The main difference among three

(a) Case 1 (b) Case 2 (c) Case 3

Fig. 7 Different stiffness cases of the frame structure

PP

EIB

EIC EIC

EIC=EIB

PP

EIB

EIC EIC

EIC/100=EIB

P

EIB

EIC EIC

100EIC=EIB

P

91


cases are changing values of the beam bending stiffness EI: i) the same bending stiffness EI in

columns and beam, ii) the beam bending stiffness is 1/100 of the bending stiffness for columns, iii)

the beam bending stiffness 100 times bigger than the bending stiffness for columns.

The corresponding analytic solution for each case 2 or 3 can be easily identified from the semi-

analytic solution frame value defined in Eq. (55).

Case 2: In this case the contribution of the beam to column tip rotation is very small. In order to

define the behavior of the columns, we can neglect the contribution of the beam. Therefore, the value

of the critical load tends to the value we already computed for a single cantilever 2 22 05crP . EI l= .

Case 3: In this case the stiffness of the beam is so dominant over the stiffness of the columns that

it does not allow any rotation at the top of the columns. This results with the follower load acting as

a conservative load. Therefore, the solution is ( )22 2crP EI l=

3. Weak form based solution by finite element method

Since the follower force instability of cantilever cannot be solved using statics approach (Bolotin

1963), we here construct the corresponding dynamics framework. In particular, in this section we

will show how such dynamics framework can be generalized to more complex problems that can be

solved by using the finite element method. In constructing the weak form solution, the key role is

played by von Karman type of strain measure. In our previous work (Ibrahimbegovic et al. 2013,

Hajdo et al. 2020), the von Karman strain was used for formulating instability problems as linear

buckling analysis of structures, which reduces to solving the linear eigenvalue problem. Here, we

carry on further to introduce the Euler-Bernoulli beam with the von Karman strain measure, as the

model capable of accounting for geometric nonlinearities. We will also add the special finite element

that can take into account the follower force as the source of instability.

3.1 Variational formulation with von Karman strain as generalization of Euler buckling approach

If we consider for example an axially compressed beam, the strong form of the problem can be

written for the case of small deformations and moderate rotations as

4 2

4 20

d v P d v

dx EI dx+ = (57)

where v is transverse displacement, P is compressive force, E is Young’s modulus and I is moment

of inertia of beam’s cross section. By solving this differential equation, we obtain the Euler critical

buckling force. Unfortunately, such a result can only be obtained for simple problems; if one wants

to solve more complex problems, we have to employ the weak form, or virtual work (e.g., Ibrahimbegovic 2009). The key role here is played by virtual von Karman strain constructed under

hypothesis that deformations are small and rotations are moderate. Under such hypothesis the von

Karman deformation measure can be defined for the Euler-Bernoulli beam as follows

2 2

2

1

2

vK du dv d vy

dx dx dx

= + −

(58)

92


By introducing virtual displacements (�̂�, 𝑣) to produce the perturbed configuration of the beam

deformed configuration, we can apply the Gâteaux derivative in the direction of these virtual

displacements to obtain the virtual von Karman deformation

(59)

With this result in hand we can further write the virtual work of internal forces

(60)

The beam stress resultants in terms of axial force N and bending moment M are given as

( ) ( )2

2

du d vN x EA ; M x EI

dx dx= = (61)

We note that the values of stress resultants N and M are obtained from linear theory. Further, we

can write an explicit form of the principle of the virtual work or the variational equation

∫𝑑�̂�

𝑑𝑥𝐸𝐴

𝑑𝑢

𝑑𝑥𝑑𝑥 + ∫

𝑑2𝑣

𝑑𝑥2𝐸𝐼

𝑑2𝑣

𝑑𝑥2𝑑𝑥 + ∫

𝑑𝑣

𝑑𝑥

𝑑𝑣

𝑑𝑥𝑁𝑑𝑥 = �̂�𝑓𝑒𝑥𝑡

𝑙𝑙𝑙

(62)

where fext is an external force. The latter is the basis for the approximate solution of instability

problem, which can be obtained by the finite element method (Ibrahimbegovic 2009), for a structure

of arbitrary complexity.

3.2 Finite element discrete approximation for thin beam and for thick beams

To define the finite element approximation, we will first use a two-node Euler-Bernoulli beam

element. In 2D case, each node has three degrees of freedom, two translations and one rotation. The

nodal values of the real and virtual displacements are denoted as

𝐝 = [𝑢1 𝑣1 𝜑1 𝑢2 𝑣2 𝜑2]

�̂� = [�̂�1 𝑣1 �̂�1 �̂�2 𝑣2 �̂�2] (63)

The real displacement field interpolation is given as

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 2 2

1 1 2 1 3 2 4 4

u x N x u N x u

v x H x v H x H x v H x

= +

= + + + (64)

In Eq. (64) above, Ni are the linear shape functions, and Hj are the Hermite cubic polynomials

( ) ( )

( ) ( )

( ) ( )

1 2

2 3 2 3

1 2 2

2 3 2 3

3 4 2

1

1 3 2 2

3 2

x xN x N x

l l

x x x xH x H x x

l l l l

x x x xH x H x

l l l l

= − =

= − + = − +

= − = − +

(65)

93


where l is the length of the beam element. The same kind of interpolation given in Eq. (64) is chosen

for the virtual displacement field. The derivatives of the shape functions, and the corresponding

deformations can then easily be obtained as follows

( ) ( )2 2

1 1

1 20 0 0 0

T Ta aa a

a a

ˆdN dNdu du ˆˆx u ; x u ;dx dx dx dx

dN dN

dx dx

= =

= = = =

=

Nd d N

N

(66)

We can further obtain the first and the second derivative of the transverse displacement

( ) ( )

31 2 40 0

T Tˆdv dv ˆx ; x ;dx dx

dHdH dH dH

dx dx dx dx

= =

=

Hd d H

H

(67)

( ) ( )2 2

2 2

22 2 2

31 2 4

2 2 2 20 0

T Tˆd v d v ˆx ; x ;dx dx

d Hd H d H d H

dx dx dx dx

= =

=

Bd d B

B

(68)

By replacing these results into the relation given in Eq. (62), we can finally obtain the beam

internal force as well as the material and the geometric part of the stiffness matrix.

(69)

We note that the tangent stiffness matrix consists of material and geometric matrices

( )t m g

T T T

m g

l l

;

EA EI dx ; Ndx

= +

= + =

K K K

K N N B B K H H (70)

To take into account the shear deformation, we modify the material stiffness matrix, as shown in

our previous work (Medic et al. 2013). More precisely, the entries of the material stiffness matrix

are multiplied using shear stiffness proportional coefficients

22 25 52 55

23 26 32 62 53 56 35 65

33 66

36 63

1

1

1

1

1 0 25

1

1 0 50

1

k k k k ;

k k k k k k k k ;

.k k ;

.k k ;

= − = − = − =+

= = = = = = = =+

+ = =

+

− = =

+

(71)

with ϕ as shear/bending stiffness ratio defined as

94


2

12EI

kGAl = (72)

where A is the cross-section area, I moment of inertia, l is element length, E elasticity modulus, G

shear modulus and k is shear correction factor. An infinite shear correction factor implies negligible

transverse shear deformation, and the element recovers the results in agreement with the Euler-

Bernoulli beam theory.

Modeling beam dynamics has been subject of many studies (Culver et al. 2019). In case when

applied load varies in time, the resulting displacements also vary in time. We assume that the nodal

velocities and accelerations are approximated in the same way as the nodal displacements in Eq.

(64).

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 2 2

1 1 2 1 3 2 4 4

u x,t N x u t N x u t x t

v x,t H x v t H x t H x v t H x t x t

= + =

= + + + =

N d

H d (73)

where N(x) and H(x) are the same shape functions as already used in statics

1 2

1 2 3 4

0 0 0 0

0 0

N N

H H H H

=

=

N

H (74)

The equilibrium equation in this case is replaced by equation of motion that is given as

( t )+ =Md Kd F (75)

Here, the main novelty is the mass matrix that can be obtained as follows

( )0

l

T TM A dx= + N N H H (76)

where ρ is the mass density.

3.3 Follower force finite element for thin beams

The virtual work of external forces remains as simple as indicated in Eq. (62) when fixed force

is applied. However, we have to deal with a special load case described as a follower force. The

follower force is tied to a particular cross-section (Ibrahimbegovic and Taylor 2002). The

contribution of the follower force applied at the node a to the virtual work principle is

0

ext

ext a a a tG = =f p (77)

where p0 is the initial value of the follower force, and t is the rotation matrix. The rotation matrix

in case of the Euler-Bernoulli beam that takes into account von Karman strain measure has the

following form

1 0

1 0

0 0 1

t

dv

dx

dv

dx

−

=

(78)

95


(a) Vertical force - time diagram (b) Transverse displacement vertical force diagram

indicating instability

Fig. 8 Cantilever instability under follower force

Therefore, the corresponding contribution to the beam element tangent stiffness matrix can be

obtained as

2

1

0 0

0 0

0 0 0

f

p

p

−

=

K (79)

We note that such contribution to the tangent stiffness has to be applied only to a particular beam

element node where the follower force is applied.

4. Numerical examples

4.1 Analysis of a cantilever beam

4.1.1 Cantilever beam under follower force In first example we study a cantilever beam under follower force P (see Fig. 4). The analytic

solution of such problem is presented in Section 2. Hence, this example can provide the validation

of our numerical approach.

At the top of the cantilever, we also apply a small horizontal perturbation force of 0.01P. The

chosen material and geometrical properties of the cantilever are as follows: E=106 N/cm2, cross-

section area A=1.0cm2, moment of inertia I=0.001 cm4, and length is l=100.0 cm. By replacing these

values into the analytical solution of the problem, we obtain the critical force value Pcr=2.02 N.

We here perform numerical analysis for different number of elements, from 10 to 100. The

structure is subjected to the follower force P, and at time of 1 second a small disturbing force is

applied to the top of the cantilever. The obtained results are given in Fig. 8.

The analysis performed using our model indeed results with instability phenomena for the

analytical value of critical force - 2.02 N. Namely, we can see in Fig. 9, the disproportionally large

vibration amplitudes for a small perturbation. The closer we get to analytic solution by using a mesh

with 100 elements the instability becomes more visible.

96


Fig. 9 Cantilever under follower force - time vs transverse displacement

(a) k=2.0 (b) k=0.002

(c) k=0.0002 (d) k=0.00002

Fig. 10 Shear deformation impact to the loss of stability

In Fig. 9, it is shown how dynamic instability develops in time, and how transverse displacement

at the free end of the cantilever is increasing in time for a small perturbation introduced in time t=1s.

First, we see that the disproportionally large vibration takes some time to develop. We also see that

to obtain more accurate results, we have to use a refined mesh of 20 or more beam elements.

4.1.2 Cantilever beam under follower force including shear deformation Here, we will show how shear deformation affects the value of the critical force. The analytic

97


solution is already given in Section 2.2. We analyze the same kind of cantilever as in the first

example, but with possibility to include shear deformation by using the Timoshenko beam model.

We take all the same properties, and also the shear modulus equal to G=0.5×E. In Fig. 10, we give

the results for four different values of the shear correction coefficient.

We perform analysis with different values for critical load. We can see in Fig. 10 that the higher

is the critical load value, the more quickly the instability is manifested. We also see that the presence

of shear deformation reduced the value of critical load.

We have already stated that the infinite shear correction factor implies negligible transverse shear

deformation, and the obtained results in this case match the Euler-Bernoulli beam results. Fig. 10(a)

shows how transverse displacements changes in time for different force values and shear correction

factor 2.0. These results are close to the results given in Fig. 9. If we keep increasing the value of

the correction factor, these results would match completely. But in cases when shear deformation is

not negligible, Figs. 10(b)-(d), the consequence is an increase in transverse displacement for the

same critical force value, or an increase in force value which causes the instability.

4.2 Stability analysis of a frame structure

In this section, we analyze the behavior of the frame structure (see Fig. 5) under two vertical

(a) Ibeam=100×Icolumn (b) IBeam=Icolumn

(c) IBeam=0.01×Icolumn

Fig. 11 Frame instability under conservative force

98


forces P, also loaded with small disturbances in horizontal direction. The horizontal perturbating

forces 0.01P are applied at the top of the columns. Frame span and height are both equal to l=h=100

cm. Moment of inertia of both columns is I=0.001 cm4. We perform several analyses varying the

beam cross-section properties, while column properties remain constant.

First, we analyze the behavior of the frame under conservative forces, and then we study

instability of the frame under nonconservative forces.

4.2.1 Frame under conservative load In this case we apply two vertical forces, as shown in Fig. 5, and then in time of 1.0 second we

apply small horizontal disturbing forces.

The analysis is performed for three cases of beam stiffness: beam moment of inertia equals to

columns moment of inertia Ibeam=Icolumn, beam stiffness is 100 times bigger than stiffness of columns

Ibeam=100Icolumn, and finally third analysis is carried out for the case of small beam stiffness

Ibeam=0.01Icolumn. In order to obtain the critical force value for all three cases mentioned above, we

analyze the stability of the frame due to several different values of vertical forces. The resulting

curves are given in Fig. 11. We already gave semi-analytic solution for frame instability in Section

2.3, and now we can compare those results to the results we computed using our beam finite

elements. The comparison is summarized in Table 1.

Table 1 Comparison of semi-analytic and numerical solutions for frame critical force

Beam-columns stiffness ratio Semi-analytic solution Numerical solution

Ibeam = 100Icolumn 2 23 14 0 9859crP . EI l .= = 0.98 N

Ibeam = Icolumn 2 22 71 0 7344crP . EI l .= = 0.736 N

Ibeam = 0.01Icolumn 2 21 57 0 2464crP . EI l .= = 0.26 N

The results presented in Table 1 show good matching of semi-analytic and numerical solutions

for the critical force value of the frame under conservative load.

4.2.2 Frame under follower force In the third example, we analyze the same frame as in Section 4.2.1, but in this case, we perform

instability analysis under two nonconservative follower forces. We also apply small perturbation

forces at time of 1 second. Again, we perform several analyses varying the beam cross-section

properties, while column properties remain constant.

The results of the analyses are given in Fig. 12. In case when beam moment of inertia is 100

times bigger than the column’s moment of inertia, the critical force value is 1.25 N. Note that such

a result is in good agreement with the analytical solution for rigid beam (infinite moment of inertia),

which gives the value for the critical force equal to 0.98 N.

When columns and beam have the same properties, the instability occurs for the force of 1.55 N.

Finally, in case when beam is much weaker than columns, the resulting critical force has value of

2.0 N. When beam moment of inertia tends to zero, the critical force value tends to the analytical

solution of the column under follower force.

99


(a) Ibeam=100×Icolumn (b) IBeam=Icolumn

(c) IBeam=0.001×Icolumn

Fig. 12 Frame under instability under follower force

5. Conclusions

In this paper we deal with instability problems of structures under non-conservative follower

force. We demonstrate that the proposed finite element procedure is capable of providing a very

efficient analysis for this class of problems considering the structures of any complexity (all those

that can be successfully modeled with beam finite elements). We have validated the proposed

numerical approach on several examples, including some classical results. Namely, we found out

that the solution obtained in the numerical example of cantilever column under follower force shows

a very good agreement with the analytic solution for the critical force. After the shear deformation

is included, the value of the critical load is reduced, which is also in agreement with the tendencies

we should find for this kind of problems.

The stability analysis of frame under follower force indicates that our model can be used in

solving even more complex structures. The analysis is performed for several beam-column stiffness

ratios. All obtained results are in agreement with available analytic solution.

Acknowledgements

This work was supported by funding from Region HdF (AM), Ministry of Education (EH, II)

and IUF (AI). All this support is gratefully acknowledged.

100


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